Download Effect of the Orientation of a Magnetic Field on the Resistance of a

Effect of the Orientation of a Magnetic Field on the Resistance of a YBCO
Superconductor
Philip Wales
Physics Department, The College of Wooster, Wooster, Ohio 44691, USA
(Dated: December 15, 2011)
The resistance of a Yttrium Barium Copper Oxide (YBCO) superconductor in an external magnetic field was measured as a function of the angle between the current and the magnetic field. It
was expected that the resistance would be a function of the magnetic flux in the conducting planes of
YBCO. The resistance of the sample was measured and a 50mT magnet which was oriented through
the angles 0◦ to 120◦ in both directions and then both polarities relative to the face of the sample.
An attempt to fit a sine function to the data succeed for two of the four trials. Magnetic hysteresis
was clearly observed, because identical fields on the superconductor would have different affects on
the resistance. There was a trend for the increase to be most at 100◦ , which suggests that it follows
a Cosine function. Whether or not the magnetic field created a change in the resistivity of the
superconductor or created a voltage across the superconductor that was measured as a resistance
was not determined.
I.
INTRODUCTION
Superconducting materials are known to have two
properties. First, they become ideal conductors in their
superconducting state, which means they will have a resistivity of zero. Second, any external magnetic field is
expelled from the superconductor by an equal and opposite magnetic field created by the superconductor. The
second property is known as the Meissner effect and is
often demonstrated by levitating a magnet above a superconductor. [1]
The initial experiment was an attempt to recreate an
anomaly that was observed twice during a previous experiment. The anomaly was a prolonged decreased period of negative resistance that occurred between 85◦ K
and 87◦ K, which was before the YBCO superconductor began the transition from superconducting to normal
conduction at 88◦ K. (see Figure 1)
changed to the investigation of a new effect which was
observed during experimentation. Having a sufficiently
strong external magnetic field on the superconductor
sample affected the recorded resistance of the sample as
did the angle that the field was with the current direction
in the sample.
Superconductors are defined as having a resistivity of
zero when superconducting. So the notion of the superconductor ever changing its resistivity from zero in a superconducting state would be impossible. However, the
experiment measured the voltage drop across the sample and the resistance was calculated from that and the
known constant current. So it could be considered that
the magnetic field affected the voltage across the sample
instead of its resistance.
A problem faced with this experiment was magnetic
hysteresis, which is where a material that was in an external magnetic field will partially maintain the magnetic induction after the external magnetic field has been
removed. This affected the experiment because when
the magnet was rotated around the superconductor, the
superconductor would remember the previous states of
magnetic induction. This meant that after one data point
was taken the next data point would be skewed by the
residual induction from the previous point.
II.
THEORY
FIG. 1: The two trials that exhibited negative Hall resistivity
in the previous experiment.
In order to be superconducting, a superconductor’s
temperature must be below it’s critical temperature Tc .
Not only must that condition be met but it also cannot
be in a magnetic field above a critical induction, BT nor
can it carry a current above the a critical current, Jc .
BT is temperature dependent and is calculated for the
superconductors current temperature T as,
My attempts in recreating the anomaly were unsuccessful and due to time constraints the experiment was
2 T
BT = Bc 1 −
.
Tc
(1)
2
Where Bc is the critical field strength when the temperature is 0◦ K ??
The critical current is simply the current that will produce a critical magnetic field. By ampere’s law the magnetic field strength at distance a from a wire is,
B=
µo I
.
2πa
(2)
occupy are not superconducting, which does not stop
the material from superconducting because the current
can travel around the vortices. Once the vortex lattice
becomes too crowded, there will be no superconducting
path for the current to follow and the superconductor will
stop superconducting. At this point the superconductor
has reached its upper critical limit (Hc2 ); its lower limit
(Hc1 ) is defined as when the vortices form.[7] In order
So it follows that the current which would produce a
critical magnetic field is,
Ic =
2πaBc
.
µo
(3)
Magnetic Flux is the amount of magnetic field passing
through a surface. For a flat surface, the magnetic flux
Φ is given by,
~ ·A
~
Φ=B
(4)
~ is the normal vector to the plane. This can also
Where A
be expressed as,
Φ = |B||A| cos(θ).
(5)
Where θ is the angle between the magnetic field and the
normal vector to the surface.
When a material is placed in a magnetic field, charges
near the surface are magnetically induced by the change
in magnetic flux forming diamagnetic currents that repel the magnetic field. The internal resistance of most
materials is nonzero, so the diamagnetic currents are not
enough to produce a magnetic field strong enough to fully
repel the external magnetic field. As a result, the external magnetic field penetrates the material. [6]
With a superconducting material, the internal resistance is zero so the diamagnetic currents produced form a
magnetic field that perfectly repels the external magnetic
field. Therefore, the magnetic field inside the conductor
(B) is zero. Since B is given by,
B = µµo H.
(6)
Since µo and H are not zero, then the magnetic permeability µ of a superconducting material is zero. The
complete repulsion of the magnetic field from the superconductor is known as the Meisner effect.
For type-I superconductors if the magnetic field
reaches HT then the magnetic field will enter the superconductor and it will stop superconducting.
Type-II superconductors, like YBCO, will repel an
external magnetic field as a Type-I would. However,
when the magnetic field becomes too strong and enters
the superconductor, type-II superconductors will localize the magnetic field into vortices. These vortices are
columns of diamagnetic currents surrounding the magnetic field.?? Vortices are pinned to deformations in the
lattice (Yttrium in the case of YBCO) and will also space
themselves apart evenly. The space that the vortices
FIG. 2: Magnetic field lines penetrating a cylindrical type-II
superconductor in localized vortices.
to create Negative Hall Resistivity, there must be a gap
between the vortices and the anti-vortices.[2] For YBCO
Hc1 is 5 × 10−3 Tesla at 77◦ K when the magnetic field
is perpendicular to the current, which is when vortices
form.[5] The anti-vortices are essentially vortices in the
opposite direction of normal vortices. This makes antivortices paramagnetic in nature while normal vortices are
diamagnetic.[4] Since I could not find any reference as to
how anti-vortices are formed, and I could not observe
it again, I forfeited the attempt to create negative Hall
resistivity.
III.
PROCEDURE
The YBCO sample was fitted in a brass socket that integrated the sample into a circuit. The circuit had a DC
power supply run a constant current of 100mA through
the YBCO sample and a 5Ω resistor in series. A Keithley
2010 Multimeter measured the voltage across the 5Ω resistor and the voltage across the sample. There was also
a thermocouple with one junction attached to the brass
socket and another was left at room temperature.
FIG. 3: Schematic of the circuit. The multimeter would measure the voltages across the 5Ω resistor, the sample and the
thermocouple.
3
The multimeter repeatedly measured the voltages
across the sample, the resistor and the thermocouplein sequence and transferred the values to a computer running
LabView that calculated and recorded the resistance of
the YBCO sample using the voltage across the 5Ω resistor
to calculate the current, the voltage across the sample to
determine it’s resistance as R = VI , and the temperature
using a polynomial fit created using the thermocouple
data from the charts.
Another attempt used a large solid steel cylinder inside both of the solenoids to create the largest electromagnet available to me. The electromagnet was placed
perpendicular to the current of the sample and produced
a magnetic field of 6.8 × 10−3 T. The magnetic field created was above Hc1 but at 2.5cm the magnetic field was
not enough.
A dewar containing liquid nitrogen and the sample was
on top of a protractor taped to the table. The YBCO
sample was suspended so that it was directly over the
center of the protractor and was parallel to zero degrees
on the protractor and 2 cm outside wall of the dewar, as
shown in figure 4.
five more points were taken. This process was repeated
to 120◦ . The whole trial was repeated with the magnet
initially being at 120◦ and increments moving towards
0◦ and both of those trials were repeated with the south
pole facing the superconductor instead.
In order to counteract the affects of magnetic hysteresis, after each trial, the current was turned off and the
magnet was placed in the initial position of next trial.
Then the current was turned on and the program was
started again to wait for the resistance to stabilize as
mentioned before. However, this was not done for each
point during a trial.
IV.
ANALYSIS
The averages and standard deviation of the resistances
from each 25 point set were calculated and those values
were plotted in IgorPro. Weighted fits following the equation y0 + A sin(f · x + φ) were used on each, where x was
the angle measured in radians. The trial which took data
from 120◦ to 0◦ with the south pole facing the sample
showed significant outliers in the points at 0◦ , 110◦ , 120◦
and those points were not included in the fit.
For the trials with initial positions with the north pole
at at 0◦ and with the south pole at 120◦ , the fit function
would not converge after 40 iterations. The strong deviation at the endpoints for the south pole might be due to
experimental error, because the resistance did change as
a function of the distance between the magnet and the
sample. If, for those outlier points, the magnet was not
at the same radius from the sample as it was for the other
points then there would be a strong deviation from the
expected values.
FIG. 4: Schematic of the experiment. The angle labeled
“Theta” is θ from 2. X is the angle measured.
The DC power supply was turned on and was producing a constant current of 100mA. The YBCO sample was
positioned as aforementioned, and liquid nitrogen was
poured into the dewar using a thermos. A neodymium
magnet that created a field of 50mT at the poles, was
raised 3 cm. The field strength of the magnet was measured using a gauss-meter. The mounted magnet was
placed at the edge of the protractor at 7.5 cm away from
the sample and produced a 5 × 10−3 T field where the
sample was inside the dewar at 0◦ with the North pole
of the magnet facing the sample in the dewar. Once the
resistivity stabilized, which was usually after two hundred points, the program was run again for twenty-five
points. Then the magnet was moved to 10◦ and twenty-
FIG. 5: There are only two points of intersection. The trials
with the north pole facing the sample intersect at 100◦ and
the trials with the south pole facing the sample intersect at
20◦ .
As expected, there was general trend for the measured
resistance to be greatest around where the orientation of
the magnetic field is perpendicular to the current, and
least when the magnetic field is parallel to the current.
Since YBCO conducts in lattice planes, the effect can be
4
explained as proportional to the magnetic flux through
those planes.
TABLE I: Comparison of the calculated parameters of the
trials.
Parameter origin at 120◦ North
origin at 0◦ South
yo
8.77 × 10−6 ± 1.1 × 10−6 −1.12 × 10−5 ± 5.5 × 10−7
A
3.13 × 10−5 ± 1.2 × 10−6 2.21 × 10−5 ± 5.1 × 10−7
f
1.44 ± 0.08
1.92 ± 0.07
φ
−1.80 ± 0.08
−1.82 ± 0.09
For the converging trials (0◦ S and 120◦ N) the yo parameter is incomparable. However, the other parameters
are consistent especially the phase shift, φ, that is nearly
identical at -1.81 radians, which is a phase shift of 100◦ .
Since the fit was a sine function, the phase shift was expected to be 90◦ to make a Cosine function. The extra
10◦ could be because the lattice planes are not parallel
to the length of the sample. If my hypothesis that the resistance is a function of the flux through the conducting
planes is true, then the affect could be used to measure
the orientation of the planes of a YBCO crystal.
Interestingly, for trials of the same polarity, the resistances measured for the same angle do not match. While
the effect can be explain as a consequence of magnetic
hysteresis , the resistances do not behave the same way
as the magnetic induction of a material would. For magnetic hysteresis, the absolute change in internal magnetic
induction is initially less but then increases until it begins
to approach unity with the external magnetic field. Then
the absolute change in magnetic induction decreases. For
the trials with north polarity, the absolute change in resistance was greater when the external magnetic field was
closest to the origin. This implies that the resistance
does not have a linear dependance on the orientation of
[1] Turton, Richard J. The Physics of Solids Chapter 9 (Oxford University Press, 2000).
[2] Jensen, H. J., Minnhagen, P., Sonin, E., & Weber, H. Vortex fluctuations, negative Hall effect, and thermally activated resistivity in layered and thin-film superconductors in
an external magnetic field. Europhysics Letters, 20, 463469. (1992).
[3] Cardarelli, F. Materials handbook: a concise desktop reference (second ed., pp. 477-485). (London: Springer).
(2007).
[4] Klein, U Antivortices due to competing orbital and paramagnetic pair-breaking effects Paramana journal of physics,
the magnetic field.
The experiment might have been more revealing if a
full sweep of 0◦ to 360◦ could be taken. This would have
been possible to do if the radius of the dewar was smaller.
However, because the dewar was so large, 10.5cm, the
magnet could not produce Hc1 at that distance, so the
sample was suspended off center from the dewar. This
prevented the magnet from sweeping through all possible
angles.
V.
CONCLUSION
The initial experiment was to recreate negative hall
resistivity in YBCO. During this experiment, it was observed that the orientation of a magnetic field had an effect on the measured resistance of the superconductor in
a superconducting state. The experiment was changed
to explore this affect. Due to magnetic hysteresis and
limited angles of magnetic orientation to measure, no
conclusive model could be determined. However, it was
observed that the measured resistance was greatest when
the magnetic field was perpendicular to the current and
least when it was parallel to the current. This was in
accordance with the hypothesis that the resistivity increased with an increase of magnetic flux through the
conducting planes of the YBCO lattice. If this theory is
true then the effect can be used to determine the orientation of the lattice for a YBCO crystal. Whether the
change in resistance was cause by an actual change in the
resistivity of the superconductor or from a voltage being
created across the superconductor was never determined.
Perhaps, if multiple full sweeps of the all 360◦ for each
polarity and directions was done then more conclusive
data would be gathered.
66(1) 209-217
[5] (2006) Ruixing Liang, P. Dosangh, D. A. Bonn, and
W. N. Hardy lower critical fields in and ellipsoid-shaped
YBaCuO single crystal Physical Review B, 50
[6] (1994) V.L. Ginzburg, E. A. Andryushin, Superconductivity Revised edition (World Scientific Publishing Co. Pte.
Ltd. (2004)
[7] Guy Deutscher New Superconductors, from Granular to
High Tc (World Scientific Publishing Co. Pte. Ltd.) Chapter 6 (2006)